On Pathwise Uniform Approximation of Processes with C\`adl\`ag Trajectories by Processes with Minimal Total Variation

TL;DR

This paper develops a method to find the minimal total variation cadlag function that approximates a given cadlag process within a specified accuracy, extending classical decomposition techniques to processes with infinite total variation.

Contribution

It introduces a novel approach using truncated variations to approximate cadlag processes with minimal total variation, applicable even when total variation is infinite.

Findings

Explicit formulas for the minimal total variation approximation.

Full characterization of truncated variation for Brownian motion with drift.

Calculation of the Laplace transform of truncated variation.

Abstract

For a real cadlag function $f$ and positive constant $c$ we find another cadlag function, which has the smallest total variation possible among the functions uniformly approximating f with accuracy c=2. The solution is expressed with the truncated variation, upward truncated variation and downward truncated variation introduced in the papers R. \L ochowski, Truncated variation of Brownian motion with drift, Bull. Pol. Acad. Sci. Math. Vol. 56 (2008) and R. \L ochowski, Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift - their characteristics and applications Stochastic Processes and their Applications 121 (2011). They are analogs of Hahn-Jordan decomposition of a cadlag function with finite total variation but are always finite even if the total variation is infinite. We apply obtained results to general stochastic processes with c?adl?ag trajectories and in the special case of Brownian motion with drift we apply them to obtain full characterisation of its truncated variation by calculating its Laplace transform. We also calculate covariance of upward and downward truncated variations of Brownian motion with drift. Keywords: total variation, truncated variation, uniform approximation, Brow- nian motion, Laplace transform